Dont know about that particular Suzuki. But my understanding of Suzukis is they are poser mobils rather than work horses. Therefore dont have the guts in them to pull stuff.
Having said that ive seen some more heavier duty versions floating around so maybe they have upgraded a bit. I know when i was looking for a new car we steered away from them as they were too lightweight for our needs. The one i tried was nice to drive though.
I don't know about Suzuki's but I have a similar sized vehicle that ended up not working out. Its a 2002 Jeep liberty and has a tow rate of 5000lbs (2268kg). I bought it as a interim vehicle until I found my dream truck in my price range.
However the problem isn't just with the towing capacity, you also have to factor in the length of the wheel base. In my situation, while the jeep could tow the weight of a 2 horse slant trailer plus my 2 ponies, the Jeep doesn't have a long enough wheel base to safely pull a trailer.
I had never heard about it either until I went to pick up a horse trailer from the lady I always rent from. She wouldn't let me tow with the Jeep and explained why, I double checked online and sure enough she was telling the truth.
This blurb explains it best "The wheelbase is the distance from the front of the axle to the rear axle of the tow vehicle, and very important to the control and stability. Having a longer wheelbase on your tow vehicle will give you less of a chance of the trailer weighing down the rear axle and lifting the front axle." Quoted from https://www.transwest.com/rv/blog/7...ehicle/#:~:text=3.,and lifting the front axle.